if eps < -2.4270278086576645e-11

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   31.1
2. Using strategy rm
   31.1
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   1.5
4. Applied associate--l- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
   1.5
5. Applied simplify to get
   \[\cos x \cdot \cos \varepsilon - \color{red}{\left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \cos x \cdot \cos \varepsilon - \color{blue}{(\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*}\]
   1.5
6. Using strategy rm
   1.5
7. Applied flip3-- to get
   \[\color{red}{\cos x \cdot \cos \varepsilon - (\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*} \leadsto \color{blue}{\frac{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)}^{3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot (\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)}}\]
   1.7
8. Applied simplify to get
   \[\frac{\color{red}{{\left(\cos x \cdot \cos \varepsilon\right)}^{3} - {\left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)}^{3}}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot (\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)} \leadsto \frac{\color{blue}{{\left(\cos \varepsilon \cdot \cos x\right)}^3 - {\left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)}^3}}{{\left(\cos x \cdot \cos \varepsilon\right)}^2 + \left({\left((\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)}^2 + \left(\cos x \cdot \cos \varepsilon\right) \cdot (\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*\right)}\]
   1.7

if -2.4270278086576645e-11 < eps < 1.3114572333402165e-07

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   45.2
2. Applied taylor to get
   \[\cos \left(x + \varepsilon\right) - \cos x \leadsto \frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)\]
   6.9
3. Taylor expanded around 0 to get
   \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)}\]
   6.9
4. Applied simplify to get
   \[\color{red}{\frac{1}{6} \cdot \left(\varepsilon \cdot {x}^{3}\right) - \left(\frac{1}{2} \cdot {\varepsilon}^2 + \varepsilon \cdot x\right)} \leadsto \color{blue}{\left(\varepsilon \cdot \frac{1}{6}\right) \cdot {x}^3 - \varepsilon \cdot (\frac{1}{2} * \varepsilon + x)_*}\]
   6.9

if 1.3114572333402165e-07 < eps

1. Started with
   \[\cos \left(x + \varepsilon\right) - \cos x\]
   30.7
2. Using strategy rm
   30.7
3. Applied cos-sum to get
   \[\color{red}{\cos \left(x + \varepsilon\right)} - \cos x \leadsto \color{blue}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right)} - \cos x\]
   1.2
4. Applied associate--l- to get
   \[\color{red}{\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x} \leadsto \color{blue}{\cos x \cdot \cos \varepsilon - \left(\sin x \cdot \sin \varepsilon + \cos x\right)}\]
   1.2
5. Applied simplify to get
   \[\cos x \cdot \cos \varepsilon - \color{red}{\left(\sin x \cdot \sin \varepsilon + \cos x\right)} \leadsto \cos x \cdot \cos \varepsilon - \color{blue}{(\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*}\]
   1.2
6. Using strategy rm
   1.2
7. Applied add-log-exp to get
   \[\cos x \cdot \cos \varepsilon - \color{red}{(\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*} \leadsto \cos x \cdot \cos \varepsilon - \color{blue}{\log \left(e^{(\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*}\right)}\]
   1.3
8. Applied add-log-exp to get
   \[\color{red}{\cos x \cdot \cos \varepsilon} - \log \left(e^{(\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*}\right) \leadsto \color{blue}{\log \left(e^{\cos x \cdot \cos \varepsilon}\right)} - \log \left(e^{(\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*}\right)\]
   1.5
9. Applied diff-log to get
   \[\color{red}{\log \left(e^{\cos x \cdot \cos \varepsilon}\right) - \log \left(e^{(\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*}\right)} \leadsto \color{blue}{\log \left(\frac{e^{\cos x \cdot \cos \varepsilon}}{e^{(\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*}}\right)}\]
   1.6
10. Applied simplify to get
    \[\log \color{red}{\left(\frac{e^{\cos x \cdot \cos \varepsilon}}{e^{(\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*}}\right)} \leadsto \log \color{blue}{\left(e^{\cos \varepsilon \cdot \cos x - (\left(\sin \varepsilon\right) * \left(\sin x\right) + \left(\cos x\right))_*}\right)}\]
    1.4